Theorem int0 4882

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4455	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3497	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4875	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 233	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 266	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2818	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1533   ∈ wcel 2110  ∀wral 3138  Vcvv 3494  ∅c0 4290  ∩ cint 4868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-dif 3938  df-nul 4291  df-int 4869

This theorem is referenced by:  unissint  4892  uniintsn  4905  rint0  4908  intex  5232  intnex  5233  oev2  8142  fiint  8789  elfi2  8872  fi0  8878  cardmin2  9421  00lsp  19747  cmpfi  22010  ptbasfi  22183  fbssint  22440  fclscmp  22632  rankeq1o  33627  bj-0int  34387  heibor1lem  35081